Maps preserving matrices of local reduced minimum modulus zero at a fixed vector

Abstract

Let n be an integer greater than 1, and Mn(C) be the algebra of all n×n-complex matrices. Let x0∈Cn be a nonzero vector, and Φ be a linear map on Mn(C) such that Φ(I) is invertible. For any matrix T∈Mn(C), let γ(T,x0) denote the local reduced minimum modulus of T at x0. In this paper, we show that Φ satisfiesγ(T,x0)=0⇔γ(Φ(T),x0)=0,(T∈Mn(C)), if and only if there are two invertible matrices A,B∈Mn(C) such that Ax0=A⁎x0=x0 and Φ(T)=BTA for all T∈Mn(C). When n=2, we show that the invertibility hypothesis of Φ(I) is redundant.